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Finite Size Corrections and Likelihood Ratio Fluctuations in the Spiked Wigner Model
[article]
2017
arXiv
pre-print
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Second, we prove that below the reconstruction threshold, where it becomes impossible to reconstruct the spike, the log-likelihood ratio has fluctuations of constant order and converges in distribution ...
Our model of the problem is a rank-one deformation of a Wigner matrix where the signal-to-noise ratio (SNR) is of constant order, and we are interested in the fundamental limits of detection of the spike ...
Main results
Finite size corrections to the RS formula The results we are about to present hold in a possibly slightly smaller set than D. ...
arXiv:1710.02903v1
fatcat:hu2qoid6vneghlodgc2qo742pu
Random matrix theory in statistics: A review
2014
Journal of Statistical Planning and Inference
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matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. ...
This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical ...
The research is supported by the National Science Foundation grants DMR-1035468, DMS-1106690, DMS-1209226 and DMS-1305858. ...
doi:10.1016/j.jspi.2013.09.005
fatcat:ulbutt5h7rd6pfdak4vnuxcreu
Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization
[article]
2016
arXiv
pre-print
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A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, in which a prominent eigenvector is planted into a random matrix. ...
Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike. ...
Acknowledgements The authors are indebted to Philippe Rigollet for helpful discussions and for many comments on a draft, and to Amit Singer and his group for discussions about synchronization. ...
arXiv:1609.05573v2
fatcat:buxtnl7y2zbkfipyjgl5wgj5rm
Statistical limits of spiked tensor models
[article]
2017
arXiv
pre-print
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than when the perturbation itself exceeds the bulk; we quantify the size of this effect. ...
We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. ± 1 entries, and (iii) a sparse ...
We thank Yash Deshpande and Thibault Lesieur for resolving an issue in the replica predictions section of the first version of this paper. ...
arXiv:1612.07728v2
fatcat:i5ana3ou3nbpjlf2cgqoxyk5le
Signal Processing in Large Systems: a New Paradigm
[article]
2012
arXiv
pre-print
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This results in not-so-large practical ratios n/N, sometimes even smaller than one. ...
size N, i.e. n/N→∞. ...
In the case of the spike model, it is of interest to derive the fluctuations of the extreme eigenvalues and eigenvector projections, whose limits were given in Theorem 3. ...
arXiv:1105.0060v3
fatcat:hcclx6x5qzex5obgv66qvn6dny
Spin glass to paramagnetic transition in Spherical Sherrington-Kirkpatrick model with ferromagnetic interaction
[article]
2023
arXiv
pre-print
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This paper studies spin glass to paramagnetic transition in the Spherical Sherrington-Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant J ∈ [0,1) and inverse temperature ...
The disorder of the system is represented by a general Wigner matrix. We confirm a conjecture of and that the critical window of temperatures for this transition is β = 1 + bN^-1/3√(log N) with b∈ℝ. ...
Our original motivation stems from the fact that integrals (1.1) appear in the likelihood ratio in statistical tests of spiked models in multivariate statistics. ...
arXiv:2104.07629v3
fatcat:qphvp4krgfactidiayczcrkyjm
Computation of Confidence Levels for Exclusion or Discovery of a Signal with the Method of Fractional Event Counting
[article]
2004
arXiv
pre-print
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2004 accepted arXiv:hep-ex/0405072v3 fatcat:blb6p3b3czea3h74hffhmjcngm
A method is described, which computes from an observed sample of events upper limits for production rates of particles, or, in case of appearance of a signal, the probability for an upwards fluctuation ...
It is investigated, under which conditions a Bayesian treatment of systematic errors is correct. Some numerical examples are given and compared with the results of other methods. ...
It was introduced to circumvent bad knowledge of the spectral shapes of systematic errors in regions where the difference between the models is small. ...
arXiv:hep-ex/0405072v1
fatcat:fc2nqjrhgfgkhmab5pgioaima4
Complexity and Emergence of Meaning: Toward a Semiophysics
[chapter]
2005
New Economic Windows
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Arecchi F.T. 1995 "Truth and certitude in the scientific language", International Conference on "Self-Organization of Complex Structures from Individual to Collective Dynamics", Berlin 24-28 Sept. 1995 ...
Arecchi F.T. 2000 "Complexity and adaptation: a strategy common to scientific modeling and perception" Cognitive Processing 1, 23. ...
In a similar way, a visual system (the eye, or a telecamera) frames a finite two-dimensional domain of sizes 2 1 , L L with bandwidths 1 B and 2 B . ...
doi:10.1007/978-88-470-0344-6_1
fatcat:olpibkvsjzfelfir6nhargi3xa
Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
[article]
2020
arXiv
pre-print
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Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model. ...
This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. ...
As a by-product of our analysis, we give new bounds on the low-degree likelihood ratio for a wide class of multi-spiked matrix models (both Wigner and Wishart), which may be of independent interest. ...
arXiv:2008.12237v1
fatcat:q6dhj75mhnfndfmfdr62ssdyua
A Random Matrix Theory Approach to Damping in Deep Learning
[article]
2022
arXiv
pre-print
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noise-to-signal ratio and harming generalisation. ...
We conjecture that the inherent difference in generalisation between adaptive and non-adaptive gradient methods in deep learning stems from the increased estimation noise in the flattest directions of ...
from the spiked covariance model [Baik and Silverstein, 2004] . ...
arXiv:2011.08181v5
fatcat:nyb2ckkmdndvfi4knz6ub6wqrm
Financial Applications of Random Matrix Theory: a short review
[article]
2009
arXiv
pre-print
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We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade ...
This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum and its various generalisations, random SVD, free matrices, largest eigenvalue statistics ...
Acknowledgements We want to warmly thank all our collaborators on these topics, in particular Giulio Biroli, Laurent Laloux and Augusta Miceli. ...
arXiv:0910.1205v1
fatcat:d3af4j7jh5axbdaxskrnz3xzwa
Positive P representation: Application and validity
1997
Physical Review A. Atomic, Molecular, and Optical Physics
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We explore the range of validity of the time-evolution equations both analytically and by numerical investigation of a number of specific examples. ...
We present practical ways of verifying the validity of the use of the positive P representation and find that the standard time-evolution equation can become invalid when nonlinear terms ͑at unit photon ...
the finite ensemble size. ...
doi:10.1103/physreva.55.3014
fatcat:qnstuohwvzgrnbajl5p5dzwhlu
Quasi-universality in single-cell sequencing data
[article]
2018
arXiv
pre-print
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Applications range from understanding the transcriptional and epigenetic processes involved in metazoan development to characterizing distinct cells types in heterogeneous populations like cancers or immune ...
Interestingly, 5% of the spectrum shows deviations from these distributions and present a phenomenon known as eigenvector localization, where information tightly concentrates in groups of cells. ...
ACKNOWLEDGEMENTS This work was funded in part by R01CA185486-01, R01 CA179044-01A1, U54 U54CA209997, NIH U54 CA193313 and a Chan-Zuckerberg pilot grant. ...
arXiv:1810.03602v1
fatcat:hamvedltwrayrc63hh2u4l5eay
Signal Processing in Large Systems: A New Paradigm
2013
IEEE Signal Processing Magazine
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This results in not-so-large practical ratios n/N , sometimes even smaller than one. ...
size N , i.e. n/N → ∞. ...
In the case of the spike model, it is of interest to derive the fluctuations of the extreme eigenvalues and eigenvector projections, whose limits were given in Theorem 3. ...
doi:10.1109/msp.2012.2207490
fatcat:ragdgjwxn5g2jnjxdbh4vo2lb4
Quasi-universality in single-cell sequencing data
[article]
2018
bioRxiv
pre-print
Preserved Fulltext
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Applications range from understanding the transcriptional and epigenetic processes involved in metazoan development to characterizing distinct cells types in heterogeneous populations like cancers or immune ...
Interestingly, 5% of the spectrum shows deviations from these distributions and present a phenomenon known as eigenvector localization, where information tightly concentrates in groups of cells. ...
The spike model of Johnstone provides a simple example where a finite rank perturbation is added to a large random matrix (6) . ...
doi:10.1101/426239
fatcat:a35bbzbn5vcpfopetptzstobxi
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